3.652 \(\int \frac{1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=267 \[ -\frac{b}{8 a^2 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 b \log (x) \left (a+b x^2\right )}{a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^5 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b}{4 a^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{3 a^3 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-2*b)/(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(8*a^2*(a + b*x^2)^3*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) - b/(3*a^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
- (3*b)/(4*a^4*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (a + b*x^2)/(2*a^
5*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*b*(a + b*x^2)*Log[x])/(a^6*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (5*b*(a + b*x^2)*Log[a + b*x^2])/(2*a^6*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.339114, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{b}{8 a^2 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 b \log (x) \left (a+b x^2\right )}{a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^5 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b}{4 a^4 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{3 a^3 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-2*b)/(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - b/(8*a^2*(a + b*x^2)^3*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) - b/(3*a^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
- (3*b)/(4*a^4*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (a + b*x^2)/(2*a^
5*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*b*(a + b*x^2)*Log[x])/(a^6*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (5*b*(a + b*x^2)*Log[a + b*x^2])/(2*a^6*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4])

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Rubi in Sympy [A]  time = 44.9027, size = 264, normalized size = 0.99 \[ \frac{2 a + 2 b x^{2}}{16 a x^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}} + \frac{5}{24 a^{2} x^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} + \frac{5 \left (2 a + 2 b x^{2}\right )}{24 a^{3} x^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} + \frac{5}{4 a^{4} x^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}} - \frac{5 b \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (x^{2} \right )}}{2 a^{6} \left (a + b x^{2}\right )} + \frac{5 b \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (a + b x^{2} \right )}}{2 a^{6} \left (a + b x^{2}\right )} - \frac{5 \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2 a^{6} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

(2*a + 2*b*x**2)/(16*a*x**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)) + 5/(24*a**2
*x**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)) + 5*(2*a + 2*b*x**2)/(24*a**3*x**2
*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)) + 5/(4*a**4*x**2*sqrt(a**2 + 2*a*b*x**2
+ b**2*x**4)) - 5*b*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*log(x**2)/(2*a**6*(a +
b*x**2)) + 5*b*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*log(a + b*x**2)/(2*a**6*(a +
b*x**2)) - 5*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(2*a**6*x**2)

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Mathematica [A]  time = 0.0851996, size = 119, normalized size = 0.45 \[ \frac{-a \left (12 a^4+125 a^3 b x^2+260 a^2 b^2 x^4+210 a b^3 x^6+60 b^4 x^8\right )-120 b x^2 \log (x) \left (a+b x^2\right )^4+60 b x^2 \left (a+b x^2\right )^4 \log \left (a+b x^2\right )}{24 a^6 x^2 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(-(a*(12*a^4 + 125*a^3*b*x^2 + 260*a^2*b^2*x^4 + 210*a*b^3*x^6 + 60*b^4*x^8)) -
120*b*x^2*(a + b*x^2)^4*Log[x] + 60*b*x^2*(a + b*x^2)^4*Log[a + b*x^2])/(24*a^6*
x^2*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.03, size = 219, normalized size = 0.8 \[{\frac{ \left ( 60\,\ln \left ( b{x}^{2}+a \right ){x}^{10}{b}^{5}-120\,{b}^{5}\ln \left ( x \right ){x}^{10}+240\,\ln \left ( b{x}^{2}+a \right ){x}^{8}a{b}^{4}-480\,a{b}^{4}\ln \left ( x \right ){x}^{8}-60\,a{b}^{4}{x}^{8}+360\,\ln \left ( b{x}^{2}+a \right ){x}^{6}{a}^{2}{b}^{3}-720\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{6}-210\,{a}^{2}{b}^{3}{x}^{6}+240\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{a}^{3}{b}^{2}-480\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{4}-260\,{a}^{3}{b}^{2}{x}^{4}+60\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{4}b-120\,{a}^{4}b\ln \left ( x \right ){x}^{2}-125\,{a}^{4}b{x}^{2}-12\,{a}^{5} \right ) \left ( b{x}^{2}+a \right ) }{24\,{a}^{6}{x}^{2}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24*(60*ln(b*x^2+a)*x^10*b^5-120*b^5*ln(x)*x^10+240*ln(b*x^2+a)*x^8*a*b^4-480*a
*b^4*ln(x)*x^8-60*a*b^4*x^8+360*ln(b*x^2+a)*x^6*a^2*b^3-720*a^2*b^3*ln(x)*x^6-21
0*a^2*b^3*x^6+240*ln(b*x^2+a)*x^4*a^3*b^2-480*a^3*b^2*ln(x)*x^4-260*a^3*b^2*x^4+
60*ln(b*x^2+a)*x^2*a^4*b-120*a^4*b*ln(x)*x^2-125*a^4*b*x^2-12*a^5)*(b*x^2+a)/x^2
/a^6/((b*x^2+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2688, size = 279, normalized size = 1.04 \[ -\frac{60 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} + 125 \, a^{4} b x^{2} + 12 \, a^{5} - 60 \,{\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{5} x^{10} + 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} + 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (x\right )}{24 \,{\left (a^{6} b^{4} x^{10} + 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} + 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^3),x, algorithm="fricas")

[Out]

-1/24*(60*a*b^4*x^8 + 210*a^2*b^3*x^6 + 260*a^3*b^2*x^4 + 125*a^4*b*x^2 + 12*a^5
- 60*(b^5*x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*a^3*b^2*x^4 + a^4*b*x^2)*log(b
*x^2 + a) + 120*(b^5*x^10 + 4*a*b^4*x^8 + 6*a^2*b^3*x^6 + 4*a^3*b^2*x^4 + a^4*b*
x^2)*log(x))/(a^6*b^4*x^10 + 4*a^7*b^3*x^8 + 6*a^8*b^2*x^6 + 4*a^9*b*x^4 + a^10*
x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(1/(x**3*((a + b*x**2)**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.618559, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^3),x, algorithm="giac")

[Out]

sage0*x